%I #108 Aug 16 2024 04:45:39

%S 1,3,12,57,300,1686,9912,60213,374988,2381322,15361896,100389306,

%T 663180024,4421490924,29712558576,201046204173,1368578002188,

%U 9366084668802,64403308499592,444739795023054,3082969991029800

%N Number of planar rooted trees with n nodes and tricolored end nodes.

%C Essentially the same as A025231.

%C Also number of lattice paths from (0,0) to (n-1,n-1), with steps (1,0),(0,1) and (1,1), that never rise above the line y=x and the steps (1,1) are colored red or blue. - _Emeric Deutsch_, May 28 2003

%C The Hankel transform (see A001906 for definition) of this sequence forms A049656(n+1) = [1, 3, 27, 729, 59049, 14348907, ...]. - _Philippe Deléham_, Aug 29 2006

%C With a(0)=0, this is the series reversion of x(1-x)/(1+2x). - _Paul Barry_, Oct 18 2009

%C Row sums of the Riordan matrix A121576. - _Emanuele Munarini_, May 18 2011

%D Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

%H Vincenzo Librandi, <a href="/A047891/b047891.txt">Table of n, a(n) for n = 1..200</a>

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.

%H Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry2/barry126.html">A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations</a>, J. Int. Seq. 14 (2011), Article 11.3.8.

%H Z. Chen and H. Pan, <a href="http://arxiv.org/abs/1608.02448">Identities involving weighted Catalan-Schroder and Motzkin Paths</a>, arXiv:1608.02448 [math.CO] (2016), eq. (1.13), a=3, b=1.

%H Shishuo Fu and Yaling Wang, <a href="https://arxiv.org/abs/1908.03912">Bijective recurrences concerning two Schröder triangles</a>, arXiv:1908.03912 [math.CO], 2019.

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

%H Luis Verde-Star, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>.

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F G.f.: (1 - 2*x - sqrt(1 - 8*x + 4*x^2))/2.

%F For n>0, a(n+1) = (1/n)*Sum_{k=0..n} 3^k*C(n, k)*C(n, k-1) - _Benoit Cloitre_, May 10 2003

%F a(1)=1, a(n) = 2*a(n-1) + Sum_{i=1..(n-1)} a(i)*a(n-i). - _Benoit Cloitre_, Mar 16 2004

%F The Hankel transform (see A001906 for definition) of this sequence form A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ...]. - _Philippe Deléham_, Aug 29 2006

%F 2*a(n) = A054872(n+1). - _Philippe Deléham_, Aug 17 2007

%F From _Paul Barry_, Feb 01 2009: (Start)

%F G.f.: x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction);

%F a(n+1) = Sum_{k=0..n} C(n+k,2k)*2^(n-k)*A000108(k). (End)

%F G.f.: x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-... (continued fraction). - _Paul Barry_, Oct 18 2009

%F a(1) = 1, for n>=1, a(n+1) = 3*A007564(n). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009

%F From _Emanuele Munarini_, May 18 2011: (Start)

%F a(n+1) = (Sum_{k=0..n} binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k)/2.

%F D-finite with recurrence: (n+2)*(n+3)*a(n+3) - 6*(n+2)^2*a(n+2) - 12*(n)^2*a(n+1) + 8*n*(n-1)*a(n) = 0. (End)

%F G.f.: A(x) = (1-2*x-sqrt(4*x^2-8*x+1))/2 = 1 - G(0); G(k)= 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Jan 05 2012

%F G.f.: x/W(0), where W(k)= k+1 - 2*x*(k+1) - x*(k+1)*(k+2)/W(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Aug 16 2013

%F From _Vladimir Reshetnikov_, Nov 01 2015: (Start)

%F a(n) = 2^(n-1)*(LegendreP_n(2) - LegendreP_{n-2}(2))/(2n-1).

%F a(n) = 3*hypergeom([1-n,2-n], [2], 3) - 2*0^(n-1). (End)

%F a(n) = 2^(n-1)*hypergeom([1-n, n], [2], -1/2). - _Peter Luschny_, Nov 25 2020

%F a(n) ~ 3^(1/4) * (1 + sqrt(3))^(2*n - 1) / (2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 31 2021

%F D-finite with recurrence n*a(n) +4*(-2*n+3)*a(n-1) +4*(n-3)*a(n-2)=0. - _R. J. Mathar_, Aug 01 2022

%e G.f. = x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1686*x^6 + 9912*x^7 + ...

%p A047891_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

%p for w from 1 to n do a[w] := 3*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A047891_list(20); # _Peter Luschny_, May 19 2011

%t CoefficientList[Series[(1-2x-Sqrt[1-8x+4x^2])/(2x),{x,0,100}],x] (* _Emanuele Munarini_, May 18 2011 *)

%t a[ n_] := SeriesCoefficient[(1 - 2 x - Sqrt[1 - 8 x + 4 x^2]) / 2, {x, 0, n}]; (* _Michael Somos_, Apr 10 2014 *)

%t Table[2^(n-1) (LegendreP[n, 2] - LegendreP[n-2, 2])/(2n-1), {n, 1, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)

%t Table[3 Hypergeometric2F1[1-n, 2-n, 2, 3] - 2 KroneckerDelta[n-1], {n, 1, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)

%o (PARI) a(n)=if(n<2,n==1,n--;sum(k=0,n,3^k*binomial(n,k)*binomial(n,k-1))/n)

%o (PARI) x='x+O('x^100); Vec((1-2*x-sqrt(1-8*x+4*x^2))/2) \\ _Altug Alkan_, Nov 02 2015

%o (Maxima) makelist(sum(binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k,k,0,n)/2,n,0,24); /* _Emanuele Munarini_, May 18 2011 */

%o (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-2*x-Sqrt(1-8*x+4*x^2))/(2*x))); // _G. C. Greubel_, Feb 10 2018

%Y Cf. A006318, A121576, A054872.

%K nonn,eigen,easy

%O 1,2

%A _Louis Shapiro_

%E More terms from _Christian G. Bower_, Dec 11 1999